What is the "path"?

The path is the part of the url after the "?q=" in the address bar.
To call a node page you usually put node/*nid* where *nid* is the number of the node you want to link in the menu.
Using the Path module you can create path aliases, so you can give a specific alias to your node.
For instance, if you have a page for your site infos in the node number 6, you can give this node an alias like "about", in this way your info page will be accessible with both "node/6" or "about" paths.
You can't put a menu link as "menu-www/redirection" if you previously hadn't create a path like this.

In Drupal, the path is this part that is shown in the browser's address bar:

http://drupal.org/node/363879 (with clean urls)
http://drupal.org/?q=node/363879 (no clean urls)

Contrary to what you might be inclined to think, it is not neccesarrily a path to an actual folder on the server, although it sometimes can be, especially when setting up paths to the files system in your admin settings. Most of the time though, it's the part of the url as outlined above.

Now, when you create a new node, it automatically creates a path in the form of node/#. If you enable the Path module (which is a core module), you can also create a path alias manually. In that case the node gets two paths: the node/# path and the alias you typed in. For example: http://www.example.com/node/12345 could have an alias http://www.example.com/aboutus. The main purpose of these aliases is search engine optimalization. Path aliases can be freely chosen (but an alias may not already exist) and should typically makes sense in relation to the content of the page.

So with that said, how about that menu path? Well, in short: it is the path to the node you are trying to make a menu link for. So, if you have a page with an "about us" section which can be accessed through http://www.example.com/node/123, you want to create a menu with path node/123. So, before you can create a menu link, you have to make sure the node (about us page, forum listing, and so on) exists, in order to prevent the invalid path error.

One final note. When you create a node and you have the appropriate permissions, you can also add the menu item during the creation of the node. You will find a collapsable fieldset called Menu settings. Any menu item you create during node creation will also be listed in the admin menu page.

Hope that helps. :)